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          <h1 class="post-title" itemprop="name headline">Point Set Registration: Coherent Point Drift</h1>
        

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        <p>2010年的900多引用的文章：Point Set Registration: Coherent Point Drift</p>
<p>文章里把点集配准分成rigid和nonrigid两种。rigid变换是指只包括平移，旋转，缩放的变换；nonrigid变换范围更广，其中最典型的就是仿射变换(affine),现实场景中的复杂情况也包括在这一类里。文中针对这两种情况下更健壮的配准，改进现有的算法的问题(错误提取的outlier和噪声多，容易收敛在局部最小),提出一种Coherent Point Drift(CPD)算法。<br><a id="more"></a><br>算法把点集匹配考虑成概率密度的估计问题。其中一组点作为高斯混合模型(GMM)的质心，另一组点作为数据。理想的情况下，给定的数据点的对应点由它在GMM中的后验概率的最大值得到。本文CPD的核心是GMM中的质心有一个coherent的约束来保留点集的拓扑结构。</p>
<h2 id="GMM"><a href="#GMM" class="headerlink" title="GMM"></a>GMM</h2><p>假设有两组点$X_{N\times D}=(x_1,…,x_N)^T$，$Y_{M\times D}=(y_1,…,y_M)^T$。GMM模型的概率密度函数(X点集内出现x点的概率)如下：</p>
<script type="math/tex; mode=display">{p(x)=\sum_{m=1}^{M+1}P(m)p(x|m)}</script><p>其中${p(x|m)}$为:${p(x|m)=\frac{1}{(2\pi\sigma^2)^{D/2}}exp^{-\frac{||x-y_m||^2}{2\sigma^2}}}$<br>约定在${p(x|m)}$中都使用相同的$\sigma^2$，GMM中的每个成分的${P(m)=\frac{1}{M}}$都相等。其中再加上一项平均分布${p(x|M+1)=\frac{1}{N}}$来代表噪声的影响。那么上式就变为：</p>
<script type="math/tex; mode=display">{p(x)=w\frac{1}{N}+(1-w)\sum_{m=1}^{M+1}\frac{1}{M}p(x|m)}</script><p>$y_m$和$x_n$的对应可能性表示为:</p>
<script type="math/tex; mode=display">P(m|x_n)=\frac{P(m)p(x_n|m)}{p(x_n)}</script><p>使用最大似然来估计$\theta$(Transformation变换的参数)和$\sigma^2$，最大化$L(\theta,\sigma^2)$:</p>
<script type="math/tex; mode=display">{L(\theta,\sigma^2)=\prod_{i=1}^{N}p(x_i)=\prod_{i=1}^{N}\sum_{m=1}^{M+1}P(m)p(x_i|m)}</script><p>等同于求-log最小:</p>
<script type="math/tex; mode=display">E(\theta,\sigma^2)=-\sum_{n=1}^{N}\log\sum_{m=1}^{M+1}P(m)p(x_n|m)</script><p>-log为凹函数，使用Jensen不等式求得$E(\theta,\sigma^2)$上界，再去掉常数项后的$Q(\theta,\sigma^2)$:</p>
<script type="math/tex; mode=display">Q=-\sum_{n=1}^{N}\sum_{m=1}^{M+1}P^{old}(m|x_n)\log P(x_n,m)=-\sum_{n=1}^{N}\sum_{m=1}^{M+1}P^{old}(m|x_n)\log(P^{new}(m)p^{new}(x_n|m))</script><p>问题转化为最小化成本函数$Q$。使用EM(Expectation Maximization)算法求得使上式最小的$\theta$和$\sigma^2$,在E步使用前一次的参数(首次需要初始化)算出后验概率，在M步最大化Q函数，估计新的$\theta$和$\sigma^2$。</p>
<ul>
<li>E步,根据数据点算出后验概率:<script type="math/tex; mode=display">P^{old}(m|x_n)=\frac{exp^{-\frac{1}{2}||\frac{x_n-T(y_m,\theta^{old})}{\sigma^{old}}||^2}}{\sum_{k=1}^{M}exp^{-\frac{1}{2}||\frac{x_n-T(y_k,\theta^{old})}{\sigma^{old}}||^2}+c}</script>其中$T(y,\theta)$表示对y使用以$\theta$为参数的transformation后的结果,$c=(2\pi\theta^2)^{D/2}\frac{w}{1-w}\frac{M}{N}$。</li>
<li>M步，最大化Q，估计新的$\theta$和$\sigma^2$：<script type="math/tex; mode=display">Q(\theta,\sigma^2)=\frac{1}{2\sigma^2}\sum_{n=1}^{N}\sum_{m=1}^{M}P^{old}(m|x_n)||x_n-T(y_m,\theta)||^2+\frac{N_PD}{2}\log\sigma^2</script>其中$N_P=\sum_{n=1}^{N}\sum_{m=1}^{M}P^{old}(m|x_n)$。估计新的$\theta$和$\sigma^2$对于不同种类配准有不同的要求。</li>
</ul>
<h2 id="Rigid"><a href="#Rigid" class="headerlink" title="Rigid"></a>Rigid</h2><p>rigid配准的GMM质心定义为$T(y_m;R,t,s)=sRy_m+t$，其中$R_{D\times D}$表示旋转矩阵，$t_{D\times 1}$表示平移向量，$s$表示缩放系数。算法过程如图：</p>
<img src="/2019/09/11/Point-Set-Registration-Coherent-Point-Drift/rigid.jpg" title="Rigid point set registration algorithm">
<p>由于$R$有$R^TR=I,det(R)=1$的约束，在最小化成本函数时较为复杂。其中使用推论将成本函数转化为$tr(A^TR)$形式，将求解转化为奇异值分解，具体推导省略。</p>
<h2 id="Affine"><a href="#Affine" class="headerlink" title="Affine"></a>Affine</h2><p>affine配准时没有对旋转矩阵的约束，要方便一些，质心定义为$T(y_m;R,t,s)=sBy_m+t$。其中$B_{D\times D}$是一个affine矩阵。求最小值时直接取$Q(B,t,\sigma^2)$关于$t$的偏导数并等于0，解线性方程组即可。</p>
<img src="/2019/09/11/Point-Set-Registration-Coherent-Point-Drift/affine.jpg" title="Affine point set registration algorithm">
<h2 id="Non-rigid-CPD"><a href="#Non-rigid-CPD" class="headerlink" title="Non-rigid CPD"></a>Non-rigid CPD</h2><p>文中的重点对象，Non-rigid配准中，质心定义为$T(Y,v)=Y+v(Y)$,初始位置加上一个移位函数(速度场函数)。通过正则化项来确保函数的平滑。在$Q$函数中再加上一项$\frac{\lambda}{2}\phi(v)$，转化为关于函数$v$和$\sigma^2$的函数：</p>
<script type="math/tex; mode=display">Q(v,\sigma^2)=\frac{1}{2\sigma^2}\sum_{n=1}^{N}\sum_{m=1}^{M}P^{old}(m|x_n)||x_n-(y_m+v(y_m))||^2+\frac{N_PD}{2}\log\sigma^2+\frac{\lambda}{2}\phi(v)</script><p>为了使用变分法来估计函数(最优解表示成核函数的线性组合)，需要将$\phi(v)$转变为等价的另一种形式，使用的是$v$在再生核希尔伯特空间(RKHS)的范数:</p>
<script type="math/tex; mode=display">\phi(v)=||v||_{H^m}^2=\int_{R^D}\frac{|\widetilde v(s)|^2}{\widetilde G(s)}ds</script><p>其中$\widetilde G(s)$是高斯核函数的傅里叶变换，$\widetilde v(s)$是$v(s)$的傅里叶变换。使用格林函数法求$Q(v,\sigma^2)$的欧拉—拉格朗日微分方程的数值解后，最佳移位函数表示为:</p>
<script type="math/tex; mode=display">v(z)=\sum_{m=1}^{M}w_mG(z,y_m)+\psi(z)</script><p>上式中的$\psi(z)$是零空间项，将所有$y_m$作为矩阵带入上式，得到隐含最优解的等式是:</p>
<script type="math/tex; mode=display">(G+\lambda \sigma^2d(PI)^{-1})W=d(PI)^{-1}PX-Y</script><p>其中$W_{M\times D}$是系数矩阵，$G_{M\times M}$是核矩阵，其中的元素$g_{ij}=G(y_i,y_j)=e^{\frac{1}{2}||\frac{y_i-y_j}{\beta}||^2}$，矩阵$P_{M\times N}$的元素为后验概率$p_{mn}=P^{old}(m|x_n)$。最终变换后的$y_m$的位置表示为$T=T(Y,W)=Y+GW$。算法的过程如下：</p>
<img src="/2019/09/11/Point-Set-Registration-Coherent-Point-Drift/nonrigid.jpg" title="The Coherent Point Drift algorithm for nonrigid point set registration">
<p>CPD中的有三个自由参数$w$,$\lambda$,$\beta$。$w$表示噪声的权重，$\lambda$和$\beta$都反映平滑正则化的程度。</p>
<h2 id="总结"><a href="#总结" class="headerlink" title="总结"></a>总结</h2><p>CPD在使用高斯混合模型来解决点匹配问题的基础上，加上了对变换函数$v$的平滑(motion coherence/smoothness)约束。约束实现的方法是在总的代价函数中再加入一项正则化项。通常情况下加入正则化项就是为了防止过拟合或者施加某种约束的,来惩罚不希望得到的结果。</p>
<p>在本篇CPD中：正则化项选择的是$v$在再生核希尔伯特空间(RKHS)的范数，RKHS的核函数选择了高斯核函数。约束算法的本质是使用了核方法的优化问题【<strong>核方法</strong>：要求函数$v$，选定核函数k，设定目标函数=(<strong>正则化项</strong>：该核函数k对应的RKHS的范数)+(<strong>损失函数</strong>：GMM中得到的极大似然估计的上界函数)，得到$v$=使目标函数最小的核函数k的线性组合)】。算法里并没有显式地求出匹配后的$T$函数，而是用核技巧绕过了$T$函数的具体求解，而借助$G$和$W$（将$T$表示为高斯核函数的线性组合），得到了需要的$P$矩阵。</p>
<p>在作者2007年的一篇Non-rigid point set registration:Coherent Point Drift文章(本文前身)里，正则化项$\phi(v)$采取范数形式，直觉上是想计算速度场函数的高频能量来反映平滑性，因此积分内的式子中分母使用了高斯核函数(不仅正定，对称，并且也是一个低通滤波)，并且做了傅里叶变换。包含这样想法的正则化项，恰好也等同运动一致理论(Motion Coherent Theory)里的条件，因此这个算法也就有了CPD的名字。</p>
<p>我一直想理解这个正则化项的意义，查阅了诸多博客和文章，发现除了作者自己的文章，很少有这个正则化项意义、如何实现”平滑约束”的解释，有引用的文章也基本上是在结论上做些修改，没有深入其中的原理。查的数学上的资料有偏微分方程，泛函，统计学习的一些内容。在长达一周的时间里，我所能得到的结论是这个正则化项也许可以从三方面解释：1.用机器学习中的核方法解释；2.从直觉上的速度场高频分量来解释；3.用运动一致理论MCT里的条件来解释。</p>
<h2 id="参考资料"><a href="#参考资料" class="headerlink" title="参考资料"></a>参考资料</h2><p>[EM]:<br>《统计学习方法》李航<br><a href="https://blog.csdn.net/lilynothing/article/details/67631634" target="_blank" rel="noopener">https://blog.csdn.net/lilynothing/article/details/67631634</a><br><a href="https://blog.csdn.net/u010626937/article/details/75116000" target="_blank" rel="noopener">https://blog.csdn.net/u010626937/article/details/75116000</a><br>[CPD]:<br>Point Set Registration: Coherent Point Drift<br>Non-rigid point set registration:Coherent Point Drift<br>《机器学习》周志华<br><a href="https://blog.csdn.net/xholes/article/details/84560795" target="_blank" rel="noopener">https://blog.csdn.net/xholes/article/details/84560795</a><br><a href="https://blog.csdn.net/a358463121/article/details/86358480" target="_blank" rel="noopener">https://blog.csdn.net/a358463121/article/details/86358480</a><br><a href="https://www.cnblogs.com/bigmonkey/p/9519387.html" target="_blank" rel="noopener">https://www.cnblogs.com/bigmonkey/p/9519387.html</a></p>

      
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